A002675 Numerators of coefficients for central differences M_{4}^(2*n).

1, 1, 1, 17, 31, 1, 5461, 257, 73, 1271, 60787, 241, 22369621, 617093, 49981, 16843009, 5726623061, 7957, 91625968981, 61681, 231927781, 50991843607, 499069107643, 4043309297, 1100586419201, 5664905191661, 1672180312771

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Comments

From Peter Bala, Oct 03 2019: (Start)

Numerators in the expansion of (2*sinh(x/2))^4 = x^4 + (1/6)*x^6 + (1/80)*x^8 + (17/30240)*x^10 + ....

Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^4 leads to a formula for the fourth central differences: f(x+2) - 4*f(x+1) + 6*f(x) - 4*f(x-1) + f(x-2) = (2*sinh(D/2))^4(f(x)) = D^4(f(x)) + (1/6)*D^6(f(x)) + (1/80)* D^8(f(x)) + (17/30240)*D^10(f(x)) + ..., where D denotes the differential operator d/dx. (End)

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• H. E. Salzer, Tables of coefficients for obtaining central differences from the derivatives, Journal of Mathematics and Physics (this journal is also called Studies in Applied Mathematics), 42 (1963), 162-165, plus several inserted tables.
• H. E. Salzer, Annotated scanned copy of left side of Table II.
• H. E. Salzer, Annotated scanned copy of left side of Table III
• E. W. Weisstein, Central Difference. From MathWorld--A Wolfram Web Resource.

Crossrefs

Cf. A002676 and A002677 (two different choices for denominators).

Also equals A002430/A002431.

Cf. A002671, A002672, A002673, A002674.

Programs

Maple

gf := (sinh(2*sqrt(x)) - 2*sinh(sqrt(x)))*sqrt(x):
ser := series(gf, x, 40): seq(numer(coeff(ser,x,n)), n=2..28); # Peter Luschny, Oct 05 2019

Extensions

More terms from Sean A. Irvine, Dec 20 2016